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{"name": "Solve a separable first order ODE, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except abs(a1))*sign(random(-1,1))", "description": "", "name": "c1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except abs(b1))*sign(random(-1,1))", "description": "", "name": "d1"}}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Solve a separable first order ODE, ", "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{(d1*a1-b1*c1)}/{a1+c1}+{(d1+b1)}*x/{a1+c1}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$y=$ [[0]] (Do not enter decimals in your answer.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the solution of the differential equation</p>\n  <p>\\[(\\var{a1}+x)y'=\\var{b1}+y,\\]</p>\n  <p>satisfying $y(\\var{c1})=\\var{d1}$.</p>", "tags": ["checked2015", "MAS1603", "MAS2105"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the solution of a first order separable differential equation of the form $(a+x)y'=b+y$.</p>"}, "advice": "<p>The differential equation is separable, so we can write</p>\n<p>\\[\\int{\\!\\frac{1}{\\var{b1}+y}\\,\\mathrm{d}y} = \\int{\\!\\frac{1}{\\var{a1}+x}\\,\\mathrm{d}x},\\]</p>\n<p>then</p>\n<p>\\[\\ln\\lvert\\var{b1}+y\\rvert=\\ln\\lvert\\var{a1}+x\\rvert+c,\\]</p>\n<p>so</p>\n<p>\\[y=\\simplify{A({a1}+x)-{b1}},\\]</p>\n<p>which is the general solution of the equation.</p>\n<p>Now,</p>\n<p>\\[\\var{d1}=y(\\var{c1})=\\simplify[std]{A({a1}+{c1})-{b1}},\\]</p>\n<p>so</p>\n<p>\\[A=\\simplify[std]{({d1}+{b1})/({a1}+{c1})}=\\simplify{{d1+b1}/{a1+c1}},\\]</p>\n<p>and then the full solution is</p>\n<p>\\[y=\\simplify[std]{{d1+b1}/{a1+c1}({a1}+x)-{b1}}=\\simplify{{(d1*a1-b1*c1)}/{a1+c1}+{(d1+b1)}*x/{a1+c1}}.\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}